Entropy Explained

[N]o one really knows what entropy really is, so in a debate you will always have the advantage.

– John von Neumann

So here’s a quick take on entropy.  It’s usually defined as dH/T – that is, change in heat energy (dH) divided by temperature (T).  Keep in mind a couple of things.  First, the temperature is absolute temperature, i.e., starting from absolute zero.  This means that in some sense, the temperature we’re discussing here is total average energy per unit of mass/stuff/whatever.  The second thing to keep in mind is dH is the amount of “energy in play” for a given heat transfer.  

OK, so the usual way to set up an entropy equation goes like this.  We have two “heat reservoirs” – think of them as big tanks full of hot water.  One is hotter than the other.  We're somehow going to let some small amount of heat energy (dH) flow between them.  In this simple example, it’s not enough to affect the temperature of either heat reservoir, because they’re that big.  (You can do the problem where they aren’t that big, but it takes calculus.  Not going there now.)  So we can write down the total change in entropy as:

dS = dH/T1 + (-dH)/T2

where: dS is the change in entropy, dH is the amount of energy transferred, and T1 and T2 are the temperatures of the two reservoirs.  The negative sign is on the last term because Reservoir 2 is losing dH of energy, while Reservoir 1 is gaining dH of energy.

Everyday experience tells us that heat will flow from the hotter reservoir to the colder reservoir.  So if T2 is the one losing heat here (remember that negative), it must be the hot one.  So let’s do some hand-waving math: if T2 > T1 (i.e., Reservoir 2 is hotter), then dH/T1 > dH/T2, and so the total change in entropy, dS, is always going to be greater than zero.  Another way of saying this is: entropy always Increases in a closed system.  (Closed?  Huh?  Whazzat mean?  It means in this simple example, heat can only flow between 1 and 2, it’s all insulated from the outside world, and there is no Resivoir 3.)

Here’s another way to get your mind around the concept: think of heat as money, and temperature as how rich someone is.  A $100 bill isn’t going to make much difference to Warren Buffett, but it will make a whale of a lot of difference to a starving undergrad.  So if there’s any mooching going on, who’s more likely do the mooching?  Which way will money tend to flow?

dS$ = $100/T1 - $100/T2

That’s right, T2 is “hotter” or richer, so it must be Warren Buffett’s wallet.  And once again, the entropy increases as the $100 bill changes hands.

But what IS entropy?  How can we think of it?  Statistical mechanics mathematically re-defines it as a measure of disorder.  In other words, in a closed system (here, two tanks full of hot water, insulated from the rest of the world) disorder always increases.

More on this as it relates to housework tomorrow.